Cyclic Subgroups :
If we pick some element a from a group G then we can consider the subset of all elements of G that are powers of a. This subset forms a subgroup of G and is called the cyclic subgroup generated by a. If forms a subgroup since it is
- Closed. If you multiply powers of a you end up with powers of a
- Has the identity. $a \ast a^{-1} = a^{0} = e$
For $\mathbb{Z_{30}}$( for additive operation),
$25+ 25= 50= 20 (\mod 30).$
$25+ 25+ 25= 75= 15 (\mod 30)$.
$25+ 25+ 25+ 25= 100= 10 (\mod 30)$.
$25+ 25+ 25+ 25+ 25= 125= 5 (\mod 30).$
$25+25+ 25+ 25+ 25+ 25= 150= 0 (\mod 50).$
$25+25+ 25+ 25+ 25+ 25+ 25= 175= 25 (\mod 30).$
There are $6$ such groups.
Is there any better approach ?